CN110134010A - A kind of power attraction repetitive control using equivalent disturbance compensation servo-system - Google Patents

Abstract

A kind of power attraction repetitive control using equivalent disturbance compensation servo-system, is given module and generates periodic reference signal, construct periodic feedback link, introduced equivalent disturbance compensation in power attracts and restrains, estimated using observer equivalent disturbance；Attract rule building perfect error dynamic based on power, and according to perfect error dynamic design controller, is inputted the signal being calculated as the control of servo-system；Specific attitude conirol can be carried out according to characterization system convergence performance indicator, and give the monotone decreasing region of characterization tracking error convergence process, absolute attractable layer, steady-state error first enter the calculation formula of steady-state error band maximum step number with boundary and tracking error.Power provided by the invention with equivalent disturbance compensation attracts repetitive controller, by the estimation to equivalent disturbance, can be improved systematic tracking accuracy and complete inhibition periodic perturbation.

Description

Power attraction repetitive control method adopting equivalent disturbance compensation servo system

Technical Field

The invention relates to a power attraction repetitive control method based on equivalent disturbance estimation, which is suitable for a periodic position servo system and other industrial occasions containing periodic operation processes.

Background

When the controller is designed, the internal model principle requires that the closed-loop system comprises an input signal model, namely the input signal model is implanted into the controller to form a feedback control effect, so that the output of the closed-loop system tracks the input reference signal without dead error. Repetitive control provides a controller design method based on an internal model principle, which has 'memory' and 'learning' characteristics to output an error signal to correct the control input of the previous cycle. It can completely suppress periodic disturbance, thereby realizing accurate control. The repetitive control technology is applied to high-precision servo systems such as power electronic circuits, industrial robots, hard disk drives and the like.

The attraction law method directly adopts a tracking error signal, and the controller is more direct and simpler in design. The usual attraction law reflects the desired system error dynamics when perturbations are not considered; for the situation of interference, because of the interference item, the controller designed directly according to the attraction law cannot be realized. The solution is to embed interference suppression measures into the original attraction law, construct an ideal error dynamic with disturbance suppression effect, and design a controller according to the constructed ideal error dynamic equation. Thus, the closed loop system dynamics is determined by the ideal error dynamics and has a desired tracking performance characterized by the ideal error dynamics.

For designing a digital controller by an attraction law method, designing by a discretized continuous time attraction law, and providing performance indexes describing transient and steady-state behaviors of tracking errors by analyzing ideal error dynamic characteristics, the method specifically comprises the following four indexes: an absolute attraction layer, a monotonically decreasing region, a steady state error band, and a maximum number of steps into the steady state error band. The specific values of the indexes depend on the controller parameters and the equivalent interference signal boundaries, so that the controller parameters and the equivalent interference signal boundaries are different, and the values of the three indexes are also different. Once the ideal error dynamic form is given, specific expressions of various indexes can be given in advance for guiding the parameter setting of the controller.

The Extended State Observer (ESO) is a core unit of an active disturbance rejection control system, and the basic method is to define total disturbance (including internal disturbance and external disturbance) as a new state, construct a state observer of an extended state (including total disturbance action) by using a state observation method. The method can estimate the system state, and can also estimate the real-time action quantity of overall disturbance in a system model, so as to compensate the influence of disturbance signals. Since the overall disturbance encompasses uncertainties in the system model, the system model is greatly simplified, and the control gain can also be considered known, facilitating controller design. The extended observer provides a general and practical method for observing uncertain characteristics.

Disclosure of Invention

In order to overcome the defects that the system tracking precision of the existing power attraction repetitive control method is lower and the periodic disturbance cannot be inhibited, the invention provides the power attraction repetitive control method adopting an equivalent disturbance compensation servo system, in order to enable a closed-loop system to have preset expected error tracking performance, a motor servo repetitive controller is designed according to an ideal error dynamic equation of a power attraction structure, the periodic disturbance component is completely inhibited, meanwhile, a disturbance observer is introduced into the closed-loop system to compensate the aperiodic disturbance and further improve the control performance, so that the motor servo system realizes high-speed and high-precision tracking; the invention expands the disturbance effect influencing the system output into a new variable to construct a disturbance observer, the disturbance observer does not need to directly measure a disturbance signal and know a specific model of the disturbance signal, and the invention specifically provides a specific expression of at most four indexes, namely a steady state error band, an absolute attraction layer, a monotone subtraction area and a step number required for a tracking error to enter the steady state error band for the first time, and can be used for guiding the parameter setting of the controller.

The technical scheme adopted by the invention for solving the technical problems is as follows:

a power attraction repetitive controller design method adopting an equivalent disturbance compensation servo system comprises the following steps:

step 1, giving periodic reference signals to satisfy

r_k＝r_k-N(1)

Where N is the period of the reference signal, r_kAnd r_k-NReference signals respectively representing time k and time k-N;

step 2, defining tracking error

In the formula

A₁(q^-1)＝a₁+a₂q^-1+...+a_nq^-n+1＝q(A(q^-1)-1)

A(q^-1)＝1+a₁q^-1+...+a_nq^-n

B(q^-1)＝b₀q^-1+...+b_mq^-m

Satisfy the requirement of

A(q^-1)y_k＝q^-dB(q^-1)u_k+w_k(3)

Wherein e is_k+1Represents the tracking error at time k +1, r_k+1Reference signal, y, representing the time instant k +1_k+1、y_k、y_k+1-NAnd y_k-NRepresenting the output signals at times k +1, k +1-N and k-N, respectively, u_kAnd u_k-NRepresenting the input signal at times k and k-N, w_k+1And w_k+1-NRepresenting the interfering signal at times k and k-N, respectively, d represents the delay, A (q)^-1) And B (q)^-1) Is q^-1Polynomial of (a), q^-1Denotes a one-step delay operator, n denotes A (q)^-1) M represents B (q)^-1) Order of (a)₁,...,a_n,b₀,...,b_mIs a system parameter and b₀Not equal to 0, n is more than or equal to m, d is an integer and is more than or equal to 1;

step 3. constructing equivalent disturbance

d_k＝w_k-w_k-N(4)

Where N is the period of the reference signal, d_kRepresenting the equivalent disturbance signal at time k, w_kAnd w_k-NRespectively representing interference signals at the k moment and the k-N moment;

expressing the tracking error as

e_k+1＝r_k+1-y_k+1-N+A₁(q^-1)(y_k-y_k-N)-q^-d+1B(q^-1)(u_k-u_k-N)-d_k+1(5)

Wherein d is_k+1Representing the equivalent disturbance at the moment k + 1;

step 4, designing an observer and estimating equivalent disturbance

Design observer equivalent disturbance d_k+1Observing, and compensating equivalent disturbance by the observed value, wherein two observed variables of the observer areAndare used to estimate e respectively_kAnd d_kBased on the error dynamics (equation (5)), an observer of the following form is designed

Wherein,represents the error e_k+1Is estimated by the estimation of (a) a,represents the error e_kIs estimated by the estimation of (a) a,representing equivalent perturbations, β₁Indicating observer gain coefficient with respect to error, β₂Representing the observer gain coefficient with respect to the equivalent disturbance,an estimation error representing a tracking error;

estimation error of equivalent disturbanceIs composed of

Estimation error of tracking error is

The expressions (7) and (8) are written as follows

Note the bookThe characteristic equation is

|λI-B|＝0 (10)

Namely, it is

λ²+(β₁-β₂-1)λ-β₁＝0 (11)

Thus, the characteristic root is

For parameter β₁And β₂Configured so that all feature roots are within the unit circle, then matrix B is a Schur stable matrix, and the estimation error converges asymptotically, i.e.

Step 5. construct the power law of attraction with disturbance suppression measures

Wherein rho and epsilon are both adjustable parameters,denotes an attraction index, and 0 < rho < 1, epsilon > 0,

step 6, constructing a repetitive controller with equivalent disturbance compensation

Combining equation (5) and equation (12), design a repetitive controller with equivalent disturbance compensation

Note the book

Expressing a repetitive controller as

u_k＝u_k-N+v_k(14)

Will u_kThe controller input signal as servo object can measure and obtain servo system output signal y_kFollows the reference signal r_kAnd (4) changing.

Further, an expression of four indexes, such as a steady state error band, an absolute attraction layer, a monotone decreasing area, the maximum number of steps required for the tracking error to enter the steady state error band for the first time and the like is given, and the expression is used for describing the tracking performance of the system and guiding the parameter setting of the controller, wherein the steady state error band, the absolute attraction layer, the monotone decreasing area and the maximum convergence number are defined as follows:

1) monotonous decreasing region delta_MDR: when e is_kGreater than this boundary, e_kThe same number is decreased, namely the following conditions are met:

2) absolute attraction layer Δ_AAL: absolute value of system tracking error_kIf | is greater than this boundary, its | e_kI, monotonically decreases, i.e. the condition is satisfied:

3) steady state error band Δ_SSE: when the system error once converges into the boundary, the error is stabilized in the region, that is, the following condition is satisfied:

4) maximum number of convergence stepsThe tracking error passes through at mostStep into the steady state error band.

Equivalent disturbance compensation error satisfactionThe expression of each index is as follows

Monotonous decreasing region delta_MDR

Δ_MDR＝max{Δ_MDR1,Δ_MDR2} (18)

Wherein, Delta_MDR1And Δ_MDR2Are all real and are determined by equation (19).

Absolute attraction layer Δ_AAL

Δ_AAL＝max{Δ_AAL1,Δ_AAL2} (20)

Wherein, Delta_AAL1And Δ_AAL2Are all real numbers and are determined by equation (21).

Steady state error band Δ_SSE

Δ_SSE＝max{Δ_SSE1,Δ_SSE2} (22)

Wherein, Delta_SSE1And Δ_SSE2Are all real, and are determined by equation (23);

in addition, given Δ_SSEThen, the tracking error enters the maximum number of steps of the steady state error band

Wherein e is₀In order to be the initial value of the tracking error,represents the smallest integer no less than.

Further, forTwo cases, according to the given Δ_MDR、Δ_AAL、Δ_SSEDetermining a corresponding calculation formula by the expression and the convergence step number expression;

the situation is as follows:

1) monotonous decreasing region delta_MDR

1.1) whenTime of flight

1.2) whenTime of flight

1.3) whenTime of flight

Wherein

2) Absolute attraction layer Δ_AAL

2.1) whenTime of flight

2.2) whenTime of flight

2.3) whenTime of flight

Wherein

3) Steady state error band

3.1) whenOr Δ_AAL≥δ_SSETime of flight

Δ_SSE＝Δ_AAL(31)

3.2) whenTime of flight

Wherein delta_SSEIs an equationThe root of Zhengguo.

4) Number of convergence steps

Wherein e is₀In order to be the initial value of the tracking error,represents the smallest integer no less than;

the situation is as follows:

1) monotonous decreasing region delta_MDR

1.1) whenTime of flight

1.2) whenTime of flight

1.3) whenTime of flight

Wherein

2) Absolute attraction layer Δ_AAL

2.1) whenTime of flight

2.2) whenTime of flight

2.3) whenTime of flight

Wherein

3) Steady state error band

3.1) whenOr Δ_AAL≥δ_SSETime of flight

Δ_SSE＝Δ_AAL(40)

3.2) whenTime of flight

Wherein delta_SSEIs an equationRoot of Zhengguo;

4) number of convergence steps

Wherein e is₀In order to be the initial value of the tracking error,represents the smallest integer no less than.

The technical conception of the invention is as follows: a power-suction repetitive control method using an equivalent disturbance compensation servo system is provided. According to a given reference signal and the constructed equivalent disturbance, an observer is introduced to estimate the equivalent disturbance, and interference suppression measures are embedded into a power law of attraction to form ideal error dynamics with interference suppression effect, so that a repetitive controller with equivalent disturbance compensation is designed, and rapid and high-precision tracking of the given reference signal is realized.

The invention has the following beneficial effects: the method has equivalent disturbance compensation, complete suppression of periodic disturbance, fast convergence performance and high tracking precision.

Drawings

Fig. 1 is a block diagram of an ac permanent magnet synchronous motor servo system.

FIG. 2 is a block diagram of an equivalent disturbance observer.

Fig. 3 is a block diagram of a power attraction repeat controller.

FIG. 4 is a graph of the time when a disturbance w occurs_k＝5sin(2πfkT_s) +0.15sgn (sin (2k pi/150)), the simulation results for the controller parameters with ∈ 0.1, ρ 0.3, and Δ 0.3, where Δ is plotted_MDR，Δ_AALAnd delta_SSE。

FIG. 5 is a graph of the time when a disturbance w occurs_k＝-10sin(2πfkT_s) +0.15sgn (sin (2k pi/150)), the simulation results for the controller parameters with ∈ 0.1, ρ 0.3, and Δ 0.3, where Δ is plotted_MDR，Δ_AALAnd delta_SSE。

FIG. 6 is a graph of the time when a disturbance w occurs_k＝5sin(2πfkT_s) +0.15sgn (sin (2k pi/150)), the simulation results for the controller parameters with ∈ 0.15, ρ 0.5, and Δ 0.3, where Δ is plotted_MDR，Δ_AALAnd delta_SSE。

FIG. 7 is a graph of the time when a disturbance w occurs_k＝-10sin(2πfkT_s) +0.15sgn (sin (2k pi/150)), the simulation results for the controller parameters with ∈ 0.15, ρ 0.5, and Δ 0.3, where Δ is plotted_MDR，Δ_AALAnd delta_SSE。

Fig. 8 to 11 are experimental results of the permanent magnet synchronous motor control device when the feedback controller parameter is ρ ═ 0.3 and ∈ ═ 0.1, where:

FIG. 8 is a reference position signal and an actual position signal under the action of a feedback controller based on the power law of attraction;

FIG. 9 is a control voltage signal under the action of a feedback controller based on the power law of attraction;

FIG. 10 is a graph of position error under the influence of a feedback controller based on the power law of attraction;

fig. 11 is a histogram of a position error distribution by a feedback controller based on the power law of attraction.

Fig. 12-15 show the feedback controller parameters ρ ═ 0.3, ε ═ 0.1, and the observer parameters β₁＝0.2，β₂0.5, the experimental result of the permanent magnet synchronous motor control device, wherein:

FIG. 12 is a reference position signal and an actual position signal under the action of a feedback controller based on the power law of attraction and equivalent disturbance compensation;

FIG. 13 is a control voltage signal under the action of a feedback controller based on the power law of attraction and equivalent disturbance compensation;

FIG. 14 is a position error under the action of a feedback controller based on the power law of attraction and equivalent disturbance compensation;

fig. 15 is a histogram of the position error distribution under the action of a feedback controller based on the power law of attraction and equivalent disturbance compensation.

Fig. 16 to 19 show experimental results of the permanent magnet synchronous motor control device when the repetitive controller parameter is ρ 0.3 and ∈ 0.1, where:

FIG. 16 is a reference position signal and an actual position signal under the influence of a repetitive controller based on the power law of attraction;

FIG. 17 is a control voltage signal under the action of a repetitive controller based on the power law of attraction;

FIG. 18 is a graph of position error under the influence of a repetitive controller based on the power law of attraction;

fig. 19 is a histogram of a position error distribution under the action of a repetitive controller based on the power law of attraction.

Fig. 20 to 23 show that the repetitive controller parameter ρ is 0.3, the parameter ∈ is 0.1, and the observer parameter β₁＝0.2，β₂0.5, the experimental result of the permanent magnet synchronous motor control device, wherein:

FIG. 20 is a graph of a reference position signal and an actual position signal under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation;

FIG. 21 is a control voltage signal under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation;

FIG. 22 is a position error under repetitive controller action based on the power law of attraction and equivalent disturbance compensation;

fig. 23 is a histogram of the position error distribution under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation.

Fig. 24 to 27 are experimental results of the permanent magnet synchronous motor control device when the feedback controller parameter is ρ is 0.5 and ∈ is 0.15, where:

FIG. 24 is a graph of a reference position signal and an actual position signal under the action of a feedback controller based on the power law of attraction;

FIG. 25 is a control voltage signal based on the power law of attraction under the influence of a feedback controller;

FIG. 26 is a graph showing a position error under the action of a feedback controller based on the power law of attraction;

fig. 27 is a histogram of a position error distribution by a feedback controller based on the power law of attraction.

Fig. 28 to 31 show that ρ is 0.5, and ∈ is 0.15 for the feedback controller parameters, and β for the observer parameters₁＝0.2，β₂Experimental results of the permanent magnet synchronous motor control device when 0.5, which is equivalent toThe method comprises the following steps:

FIG. 28 is a graph of the reference position signal and the actual position signal under the influence of a feedback controller based on the power law of attraction and equivalent disturbance compensation;

FIG. 29 is a control voltage signal under the action of a feedback controller based on the power law of attraction and equivalent disturbance compensation;

FIG. 30 is a graph of position error under the influence of a feedback controller based on the power law of attraction and equivalent disturbance compensation;

fig. 31 is a position error distribution histogram under the action of a feedback controller based on the power law of attraction and equivalent disturbance compensation.

Fig. 32 to 35 are experimental results of the permanent magnet synchronous motor control device when the repetitive controller parameter is ρ 0.5 and ∈ 0.15, where:

FIG. 32 is a graph of a reference position signal and an actual position signal under the influence of a repetitive controller based on the power law of attraction;

FIG. 33 is a control voltage signal under the influence of a repetitive controller based on the power law of attraction;

FIG. 34 is a graph of position error under the influence of a repetitive controller based on the power law of attraction;

fig. 35 is a histogram of the distribution of position errors under the action of a repetitive controller based on the power law of attraction.

Fig. 36 to 39 show that the repetitive controller parameter ρ is 0.5, the parameter ∈ is 0.15, and the observer parameter β₁＝0.2，β₂0.5, the experimental result of the permanent magnet synchronous motor control device, wherein:

FIG. 36 is a graph of a reference position signal and an actual position signal under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation;

FIG. 37 is a control voltage signal under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation;

FIG. 38 is a position error under repetitive controller action based on the power law of attraction and equivalent disturbance compensation;

fig. 39 is a histogram of the position error distribution under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation.

Detailed Description

The following further describes embodiments of the present invention with reference to the accompanying drawings.

Referring to fig. 1-39, a power-suction repetitive control method using an equivalent disturbance compensation servo system. Wherein, fig. 1 is a block diagram of a servo system of an alternating current permanent magnet synchronous motor; FIG. 2 is a block diagram of an equivalent disturbance observer; fig. 3 is a block diagram of a power attraction repeat controller.

A power attraction repetitive control method adopting an equivalent disturbance compensation servo system comprises the following steps:

step 1, giving a periodic reference signal, and satisfying the requirement (1);

step 2, defining a tracking error, wherein a second-order difference model of a motor servo system is (2), and a tracking error of the system is (3);

step 3, constructing an equivalent disturbance (4), and expressing a system tracking error as (5) by using the equivalent disturbance (4);

step 4, designing an observer and estimating equivalent disturbance;

step 5, constructing a power attraction law (12) with disturbance suppression measures;

and 6, constructing a repetitive controller with equivalent disturbance compensation, combining the formula (5) and the formula (12), designing a repetitive controller (13) with equivalent disturbance compensation, and expressing the repetitive controller as (14).

The above repetitive controller design is explained as follows:

1) introduction of d into power attraction law_k+1Reflecting suppression measures for disturbing signals of a given periodic pattern, introducedAn estimate of the equivalent disturbance is reflected, thereby providing equivalent disturbance compensation.

2) In the formula (13), e_k、y_k、y_k-1、y_k-1-NAll can be obtained by measurement, u_k-1、u_k-1-NThe stored value for the control signal may be read from memory.

3) When the reference signal satisfies r_k＝r_k-1The discrete repetitive controller is also suitable for the constant value regulation problem, and the equivalent disturbance is d_k＝w_k-w_k-1(ii) a Wherein r is_k-1Reference signal representing the time instant k-1, w_k-1Representing the interference signal at time k-1; the feedback controller with equivalent disturbance compensation is

4) The discrete time controller is designed for a second-order system, and the design result of a higher-order system can be given according to the same method.

Furthermore, an expression of four indexes, namely a steady-state error band, an absolute attraction layer, a monotone decreasing area and the maximum step number required for the tracking error to enter the steady-state error band for the first time is given, and the expression is used for describing the tracking performance of the system and guiding the parameter setting of the controller.

Further, forTwo cases, according to the given Δ_MDR、Δ_AAL、Δ_SSEAnd determining a corresponding calculation formula by the expression and the convergence step number expression.

In this embodiment, for example, the permanent magnet synchronous motor device executes a repetitive tracking task in a fixed interval, and the position reference signal has a periodic symmetry characteristic. TMS320F2812DSP is used as a controller, a Korean LS AC servo motor APM-SB01AGN is used as a control object, and a permanent magnet synchronous motor servo system is formed by the ELMO AC servo driver and an upper mechanism to control the position of the motor. The servo system adopts three-loop control, the current loop and speed loop controller ELMO driver provide, and the position loop is provided by DSP development board.

Obtaining a mathematical model of the servo object by parameter estimation

y_k+1-1.8949y_k+0.8949y_k-1＝1.7908u_k-0.5704u_k-1+w_k+1(43)

Wherein, y_k，u_kPosition output and control input, w, respectively, of the position servo system_kIs an interference signal.

The effectiveness of the repetitive controller given by the present invention will be illustrated in this example by numerical simulation and experimental results.

Numerical simulation this embodiment takes a sinusoidal signal as a system reference signal, and the corresponding repetitive controller expression can be written as

Given a position reference signal of r_k＝20(sin(2πfkT_s-1/2 pi) +1) in degrees (deg), frequency f 1Hz, sampling period T_s0.001s, and the sampling period N is 1000. Selecting proper disturbance amount w during simulation_kIt consists of periodic disturbances and non-periodic random disturbances.

Under the action of a repetitive controller (44), different controller parameters rho and epsilon are selected, and three boundary layers of a servo system are different. For purposes of illustrating the invention patent with respect to the monotonically decreasing region Δ_MDRAbsolute attraction layer Delta_AALAnd steady state error band Δ_SSETheoretical correctness of (1) inNumerical simulations were performed for the examples.

1) When the controller parameter ∈ is 0.1, ρ is 0.3, and Δ is 0.3, the calculation formula of the three boundary values indicates that

Δ_SSE＝Δ_AAL＝Δ_MDR＝0.7035

2) When the controller parameter ∈ is 0.15, ρ is 0.5, and Δ is 0.3, the calculation formula of the three boundary values indicates that

Δ_SSE＝Δ_AAL＝0.3823，Δ_MDR＝0.8884

The simulation results are shown in FIGS. 4-7, where FIGS. 4 and 6 are the disturbance values w_k＝5sin(2πfkT_s) Simulation result of +0.15sgn (2k pi/150)_，FIG. 5_，7 is the disturbance quantity w_k＝-10sin(2πfkT_s) +0.15sgn (2k pi/150).

The numerical results verify the monotonous reduction area delta of the tracking error of the system under the action of the repetitive controller given by the patent under the condition of a given system model, a reference signal and an interference signal_MDRAbsolute attraction layer Delta_AALAnd steady state error band Δ_SSEThe accuracy of (2).

The block diagram of the permanent magnet synchronous motor control system for the experimental verification experiment is shown in figure 1. And verifying the tracking performance of discrete repetitive control based on the power attraction law by setting different controller parameters. Given position signal r_k＝A(sin(2πfkT_s) +1), where the amplitude a is 135deg, the sampling period T_s5ms, frequency f 1Hz, toExperimental verification was performed for the examples.

The feedback controller adopted has the following form

The feedback controller based on disturbance compensation is adopted and has the following form

The adopted repetitive controller has the following form

The repetitive controller based on disturbance compensation is adopted and has the following form

1) The controller (45) is adopted, the controller parameters are rho is 0.3, epsilon is 0.1, and the experimental results are shown in figures 8-11, wherein delta is shown in the figures_SSE＝0.15deg。

2) Adopting a controller (46), wherein the parameters of the controller are rho equal to 0.3, epsilon equal to 0.1, and the parameters of the equivalent disturbance observer are β₁＝0.2，β₂The results are shown in FIGS. 12-15, where Δ is 0.5_SSE＝0.1deg。

3) The controller (47) is adopted, the controller parameters are rho is 0.3, epsilon is 0.1, and the experimental results are shown in figures 16-19, wherein delta is shown in the figures_SSE＝0.1deg。

4) Adopting a controller (48), wherein the parameters of the controller are rho equal to 0.3, epsilon equal to 0.1, and the parameters of the equivalent disturbance observer are β₁＝0.2，β₂The results are shown in FIGS. 20-23, where Δ is 0.5_SSE＝0.08deg。

5) By means of a controller(45) The controller parameters ρ is 0.5 and ∈ is 0.15, and the experimental results are shown in fig. 24 to 27, where Δ is shown in the graphs_SSE＝0.15deg。

6) Adopting a controller (46), wherein the parameters of the controller are rho equal to 0.5, epsilon equal to 0.15, and the parameters of the equivalent disturbance observer are β₁＝0.2，β₂The results are shown in FIGS. 28-31, where Δ is 0.5_SSE＝0.11deg。

7) The controller (47) is adopted, the controller parameters are rho is 0.5, epsilon is 0.15, and the experimental results are shown in figures 32-35, wherein delta is shown in the figures_SSE＝0.1deg。

8) Adopting a controller (48), wherein the parameters of the controller are rho 0.5, epsilon 0.15, and the parameters of the equivalent disturbance observer are β₁＝0.2，β₂The results are shown in FIGS. 36-39, where Δ is 0.5_SSE＝0.07deg。

The experiment result shows that the equivalent disturbance is introduced and is estimated by the equivalent disturbance observer, the compensation for the unmodeled characteristic and the external unknown disturbance of the system is provided, and the influence of the unknown disturbance on the tracking performance can be effectively inhibited; and the periodic disturbance is completely inhibited by adopting repeated control, so that the control performance of the system is further improved.

Claims (3)

1. A power attraction repetitive control method adopting an equivalent disturbance compensation servo system, wherein a controlled object is a periodic servo system, and the method is characterized by comprising the following steps:

step 1, giving periodic reference signals to satisfy

r_k＝r_k-N(1)

Where N is the period of the reference signal, r_kAnd r_k-NReference signals respectively representing time k and time k-N;

step 2, defining tracking error

In the formula

A₁(q^-1)＝a₁+a₂q^-1+…+a_nq^-n+1＝q(A(q^-1)-1)

A(q^-1)＝1+a₁q^-1+…+a_nq^-n

B(q^-1)＝b₀q^-1+…+b_mq^-m

Satisfy the requirement of

A(q^-1)y_k＝q^-dB(q^-1)u_k+w_k(3)

Wherein e is_k+1Represents the tracking error at time k +1, r_k+1Reference signal, y, representing the time instant k +1_k+1、y_k、y_k+1-NAnd y_k-NRepresenting the output signals at times k +1, k +1-N and k-N, respectively, u_kAnd u_k-NRepresenting the input signal at times k and k-N, w_k+1And w_k+1-NRepresenting the interfering signal at times k and k-N, respectively, d represents the delay, A (q)^-1) And B (q)^-1) Is q^-1Polynomial of (a), q^-1Denotes a one-step delay operator, n denotes A (q)^-1) M represents B (q)^-1) Order of (a)₁,…,a_n,b₀,…,b_mIs a system parameter and b₀Not equal to 0, n is more than or equal to m, d is an integer and is more than or equal to 1;

step 3. constructing equivalent disturbance

d_k＝w_k-w_k-N(4)

Where N is the period of the reference signal, d_kRepresenting the equivalent disturbance signal at time k, w_kAnd w_k-NRespectively representing interference signals at the k moment and the k-N moment;

expressing the tracking error as

e_k+1＝r_k+1-y_k+1-N+A₁(q^-1)(y_k-y_k-N)-q^-d+1B(q^-1)(u_k-u_k-N)-d_k+1(5)

Wherein d is_k+1Representing the equivalent disturbance at the moment k + 1;

step 4, designing an observer and estimating equivalent disturbance

Design observer equivalent disturbance d_k+1Observing and compensating equivalent disturbance by the observed value; two observed variables of the observer areAndare used to estimate e respectively_kAnd d_k(ii) a From the error dynamics (equation (5)), an observer of the following form is designed

Wherein,represents the error e_k+1Is estimated by the estimation of (a) a,represents the error e_kIs estimated by the estimation of (a) a,representing equivalent perturbations, β₁Indicating observer gain coefficient with respect to error, β₂Representing observer gain coefficients with respect to equivalent disturbances;an estimation error representing a tracking error;

estimation error of equivalent disturbanceIs composed of

Estimation error of tracking error is

The expressions (7) and (8) are written as follows

Note the bookThe characteristic equation is

|λI-B|＝0 (10)

Namely, it is

λ²+(β₁-β₂-1)λ-β₁＝0 (11)

Thus, the characteristic root isPair β₁And β₂Is configured such that all feature roots are within the unit circle;

step 5. construct the power law of attraction with disturbance suppression measures

Wherein rho and epsilon are both adjustable parameters,denotes an attraction index, and 0 < rho < 1, epsilon > 0,

step 6, constructing a repetitive controller with equivalent disturbance compensation

Combining equation (5) and equation (12) to obtain a repetitive controller with equivalent disturbance compensation

Note the book

Expressing a repetitive controller as

u_k＝u_k-N+v_k(14)

Will u_kAs input signal of controller of servo object, measuring to obtain output signal y of servo system_kFollows the reference signal r_kAnd (4) changing.

2. The method as claimed in claim 1, wherein the expressions of four indexes, namely a steady state error band, an absolute attraction layer, a monotone decreasing region and the maximum number of steps required for a tracking error to enter the steady state error band for the first time, are given for describing the tracking performance of the system and guiding the parameter setting of the controller, wherein the steady state error band, the absolute attraction layer, the monotone decreasing region and the convergence step are defined as follows:

1) monotonous decreasing region delta_MDR: when e is_kGreater than this boundary, e_kThe same number is decreased, namely the following conditions are met:

2) absolute attraction layer Δ_AAL: absolute value of system tracking error_kIf | is greater than this boundary, its | e_kI, monotonically decreases, i.e. the condition is satisfied:

3) steady state error band Δ_SSE: when the system error once converges into the boundary, the error is stabilized in the region, that is, the following condition is satisfied:

4) maximum number of convergence stepsThe tracking error passes through at mostEntering a steady state error band;

equivalent disturbance compensation error satisfactionThe expression of each index is as follows

Monotonous decreasing region delta_MDR

Δ_MDR＝max{Δ_MDR1,Δ_MDR2} (18)

Wherein, Delta_MDR1And Δ_MDR2Are all real and are determined by equation (19);

absolute attraction layer Δ_AAL

Δ_AAL＝max{Δ_AAL1,Δ_AAL2} (20)

Wherein, Delta_AAL1And Δ_AAL2Are all real numbers and are determined by equation (21);

steady state error band Δ_SSE

Δ_SSE＝max{Δ_SSE1,Δ_SSE2} (22)

Wherein, Delta_SSE1And Δ_SSE2Are all real, and are determined by equation (23);

in addition, given Δ_SSEThen, the tracking error enters the maximum number of steps of the steady state error band

Wherein e is₀In order to be the initial value of the tracking error,represents the smallest integer no less than.

3. The method of claim 2, wherein for the power-law pull-in repetitive control using an equivalent perturbation compensation servo systemAndtwo cases, according to the monotone decreasing zone Δ given_MDRAbsolute attraction layer Delta_AALSteady state error band delta_SSEAnd maximum number of convergence stepsThe expression determines a corresponding calculation formula;

the situation is as follows:

1) monotonous decreasing region delta_MDR

1.1) whenTime of flight

1.2) whenTime of flight

1.3) whenTime of flight

Wherein

2) Absolute attraction layer Δ_AAL

2.1) whenTime of flight

2.2) whenTime of flight

2.3) whenTime of flight

Wherein

3) Steady state error band

3.1) whenOr Δ_AAL≥δ_SSETime of flight

Δ_SSE＝Δ_AAL(31)

3.2) whenTime of flight

Wherein delta_SSEIs an equationRoot of Zhengguo;

4) number of convergence steps

Wherein e is₀In order to be the initial value of the tracking error,represents the smallest integer no less than;

the situation is as follows:

1) monotonous decreasing region delta_MDR

1.1) whenTime of flight

1.2) whenTime of flight

1.3) whenTime of flight

Wherein

2) Absolute attraction layer Δ_AAL

2.1) whenTime of flight

2.2) whenTime of flight

2.3) whenTime of flight

Wherein

3) Steady state error band

3.1) whenOr Δ_AAL≥δ_SSETime of flight

Δ_SSE＝Δ_AAL(40)

3.2) whenTime of flight

Wherein delta_SSEIs an equationRoot of Zhengguo;

4) number of convergence steps

Wherein e is₀In order to be the initial value of the tracking error,represents the smallest integer no less than.